University of South Carolina
Department of Physics and Astronomy
Phys 706, Thermodynamics and Statistical Physics.
Final Exam
April 27, 2011
The test is open book/notes.
You can use any materials you like.
If you need extra
paper or you forgot a calculator – please ask.
There are 3 problems, each worth 10 points. It is enough to solve one problem to get a
full credit. Choose any one you like. If you solve more, you will get extra credit. If you have
a question about the problem’s formulation – please do not hesitate to ask.
I will either
answer your question or will tell you that I cannot do it without revealing the solution.
Your solutions should have a readable form showing the flow of argument. You have to
be able to express yourself in writing. Writing skills are important for any scientist. They
are not just making your solutions look nicer. If you cannot explain yourself clearly, you
probably do not understand what you are doing.
You can still try to guess the answer for a problem. If the solution is a number or an
expression, I will accept your answer (guessing will not be accepted for problems where the
answer is “yes” or “no”). However, if no explanations are given, only correct answers will
be graded positively.
There will be no partial credit in the absence of correct, readable
explanations.
GOOD LUCK!
1
Ideal gas with gravity field
You have a cylindrical container of height
h
and bottom area
A
filled with classical, ideal,
singleatomic gas. The total number of atoms is
N
, the mass of each atom is
m
, and the
temperature is
T
. Since every molecule is attracted to the Earth with the force
mg
, there
will be more molecules at the bottom of the cylinder than at the top.
(a)[2 points] Calculate the dependence of the gas concentration
n
(
z
) as a function of
height
z
. (The uneven
n
has to create pressures which balance the gravity force acting on
each element of gas.)
(b)[8 points] In addition to the kinetic energy the gas has some potential energy in the
gravity field.
Find the total heat capacity of the gas in the cylinder.
Find the limits at
T
→ ∞
and
T
→
0.
Solution:
(a) If you consider a thin horizothal slice of gas of between the planes
z
and
z
+
dz
, the
difference between the pressures at its bottom and top has to be equal to the weight of the
gas. This gives
mgn
(
z
)
dz
=
P
(
z
)

P
(
z
+
dz
) =

dP
dz
dz
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Using
P
=
nk
B
T
we transform this to a differential equation
dn
dz
=

mg
k
B
T
n
(
z
)
The solution is
n
(
z
) =
n
(0)
e

(
mg/k
B
T
)
z
=
n
(0)
e

z/z
0
where
n
(0) is the concentration at the bottom of the cylinder and
z
0
≡
k
B
T/mg
is a
characteristic length of concentration decay.
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 Spring '11
 Staff
 Thermodynamics, Atom, Energy, kB

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